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Unformatted text preview: test whether the graph of an
equation is symmetric about the y -axis by replacing x with −x and checking to see if an equivalent
equation results. If we are graphing the equation y = f (x), substituting −x for x results in the
equation y = f (−x). In order for this equation to be equivalent to the original equation y = f (x)
we need f (−x) = f (x). In a similar fashion, we recall that to test an equation’s graph for symmetry
about the origin, we replace x and y with −x and −y , respectively. Doing this substitution in the
equation y = f (x) results in −y = f (−x). Solving the latter equation for y gives y = −f (−x). In
order for this equation to be equivalent to the original equation y = f (x) we need −f (−x) = f (x),
or, equivalently, f (−x) = −f (x). These results are summarized below.
Steps for testing if the graph of a function possesses symmetry
The graph of a function f is symmetric:
• About the y -axis if and only if f (−x) = f (x) for all x in the domain...
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